Application of the Cayley-Hamilton Theorem

Cayley-Hamilton Theorem, a basic theorem from Linear Algebra, is not a part of JEE syllabus. But this simple result comes in handy in JEE often.

According to the Cayley-Hamilton Theorem, every square matrix A satisfies its own Characteristic Equation.

For a matrix A of size n X n, the Characteristic Equation in variable is defined as,
,
where I is the identity matrix of size n X n.
So according to the Cayley-Hamilton Theorem, we have,




This can be illustrated with an example.

Consider,

Then,

From the Cayley-Hamilton Theorem,

And it can be verified that the above result is indeed correct.

Problem 1
Let

Suppose

Then determine c and d.
                                                                                                 (IIT-JEE 2005)
 Solution 1
 The given equation gives,

By applying the Cayley-Hamilton Theorem on A, we have,



Direct comparison of (1) and (2) gives c=-6 and d=11

Differentiation Under The Integral Sign

Problems related to differentiation under the integral sign are very common in IIT-JEE. We will see how to solve such problems using Leibniz's Rule given below.

Problem 1
If f(x) is differentiable and ,

find f(4/25)
                                                                                                (IIT-JEE 2004)
Solution
Differentiating both sides,

Hence, f(4/25)=2/5 for t=2/5
PS: Note that -2/5 is also a valid answer. It was not among the options for the MCQ. 

Problem 2
Solve

                                                                                                  (IIT-JEE 2007)

Solution 
This is in 0/0 form. So we can apply L'Hospital's Rule directly. Differentiating the numerator and the denominator separately, we have,